The other aspect of Kaladins character is that he tries to understand others (besides lighteyes in his darker thoughts, but those episodes are becoming less common, and less vehemently than he used to as shown by a whole platoon of lighteyes on the guard becoming "his" people to protect), it is what gets him attached to the different groups of people (and that understanding typically leads him to wanting to protect them).
Also, if memory serves, when they confronted near the end of WoR, I think Kaladin said something like "Then you are not the evil man I thought you were, you are a coward" (I will try to look it up later and edit this with the real quote). Assuming Kaladin thought of Szeth that way when he killed him in order to protect Dalinar, there isn't a strong hatred that he has to overcome to accept him. Also, Kaladins moral code (that of the windrunners, his oaths, and Syls advice) isn't one to start a fight. He was probably distrustful, and needed to see proof of Szeth being on their side (and then some for him to become Dalinars guard), Szeths actions at the battle of Thalen city provide a large chunk of proof of Szeths loyalty.
As long as Szeth can convince Kaladin that he isn't a threat Dalinar and Dalinar wants him there, Kaladin would allow it (even if he doesn't like it). While it is arguable how much he still considers himself under Dalinars command, he knows he isn't in a position to order Dalinar around (this wouldn't be as true for Jasnah since she is Queen)